%!TEX program = xelatex
%!TEX TS-program = xelatex
%!TEX encoding = UTF-8 Unicode

\documentclass[12pt,t,aspectratio=169,mathserif]{beamer}
%Other possible values are: 1610, 149, 54, 43 and 32. By default, it is to 128mm by 96mm(4:3).
%run XeLaTeX to compile.

\input{wang-slides-preamble.tex}

\begin{document}

\title{高等代数一}
\subtitle{4-习题与问答-行列式 }
%\institute{上海立信会计金融学院}
%\author{王立庆}
\author{{\ppr LQW}}
%\renewcommand{\today}{{\ppr \number\year \,年 \number\month \,月 \number\day \,日} }
\date{{\ppr 2022年9月29日} }

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}[fragile=singleslide]{3.1.1. }
\begin{frame}{内容提要 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{enumerate}

\item  行列式的几何解释：平行四边形的面积与平行六面体的体积。

\item  行列式的拉普拉斯展开：行列式可以按固定几行展开。

\item  计算 $n$ 阶行列式的 $k$ 阶子式。

\item  排列的奇偶性：对换两个数字正好改变奇偶性。

\end{enumerate}


%\vfill 

%{\color{red}本次习题由学号最后两位数字为 $\{6k-5 \mid k=1,2,3,4,5,6,7,8\}$ 的同学讲解。}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}[fragile=singleslide]{3.1.1. }
\begin{frame}{讲解本次作业的同学 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

{\small 
\begin{table}[ht]
\centering
\begin{tabular}{cccccc}
4-习题&8-习题&12-习题&16-习题&20-习题&24-习题 \\ \hline 
\underline{01}&02&03&04&05&06 \\   
\underline{07}&08&09&10&11&12 \\  
\underline{13}&14&15&16&17&18 \\ 
\underline{19}&20&21&22&23&24 \\  
\underline{25}&26&27&28&29&30 \\  
\underline{31}&32&33&34&35&36 \\  
\underline{37}&38&39&40&41&42 \\  
\underline{43}&44&45&46&47&48 \\ 
49&50&51&52&53&54 \\  
\end{tabular}
\end{table}
}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}[fragile=singleslide]{3.1.1. }
\begin{frame}{4.1. 习题1 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  习题1：在平面直角坐标系中，记 $O=(0,0)$ 为原点。画出向量 $\overset{\longrightarrow}{OA}=(a,b)$, $\overset{\longrightarrow}{OB}=(c,d)$. 以 $\overset{\longrightarrow}{OA},\overset{\longrightarrow}{OB}$ 为边，画出平行四边形 $OACB$. 
\begin{enumerate}
\item  证明：这个平行四边形的面积为行列式 $\begin{vmatrix} a & c \\ b & d \end{vmatrix}$ 的值。
\item  如果这个行列式的值为负数，这意味着什么？
\end{enumerate}

\item  解答思路：
\begin{itemize}
\item  先考虑 $b=0,c=0$ 的情形。
\item  再考虑 $b=0$ 的情形。
\item  最后考虑一般的情形。
\end{itemize}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}[fragile=singleslide]{3.1.1. }
\begin{frame}{4.2. 相同面积的平行四边形 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{figure}
\centering
\includegraphics[height=0.7\textheight, width=0.6\textwidth]{4-det-parallelogram.png}
% \caption{ }
\end{figure}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}{3.1.1. }
\begin{frame}[fragile=singleslide]{4.3. 上一页图形的 {\ppr Python} 代码 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{python}
import numpy as np
import matplotlib.pyplot as plt

O=np.array([0,0])
A=np.array([3,1])
B=np.array([1,4])
C=A+B
D=np.array([0,B[1]-B[0]*A[1]/A[0]])
E=A+D

fig=plt.figure()
ax=fig.add_subplot(111)

ax.hlines(y=0,xmin=-1,xmax=5)
ax.vlines(x=0,ymin=-1,ymax=6)
\end{python}


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}{3.1.1. }
\begin{frame}[fragile=singleslide]{4.4.  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

{\small
\begin{python}
ax.plot( [O[0],A[0]], [O[1],A[1]], 'bo-' )
ax.plot( [O[0],B[0]], [O[1],B[1]], 'bo-' )
ax.plot( [A[0],C[0]], [A[1],C[1]], 'bo-' )
ax.plot( [B[0],C[0]], [B[1],C[1]], 'bo-' )
ax.plot( [B[0],D[0]], [B[1],D[1]], 'ro-' )
ax.plot( [A[0],E[0]], [A[1],E[1]], 'ro-' )

ax.annotate('A',xy=(3.1,0.8))
ax.annotate('B',xy=(B[0]+0,B[1]+0.3))
ax.annotate('C',xy=(C[0]+0,C[1]+0.3))
ax.annotate('D',xy=(D[0]+0.1,D[1]+0.3))
ax.annotate('E',xy=(E[0]+0,E[1]+0.3))
ax.annotate('O',xy=(O[0]-0.2,O[1]-0.4))

ax.set_xlabel('x')
ax.set_ylabel('y')
\end{python}
}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}[fragile=singleslide]{3.1.1. }
\begin{frame}{4.5. 习题2 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  习题2：在平面直角坐标系中，记 $O=(0,0)$ 为原点。画出向量 $\overset{\longrightarrow}{OA}=(a,b)$, $\overset{\longrightarrow}{OB}=(c,d)$. 则这两个向量在一条直线上，当且仅当 
$$\begin{vmatrix} a & c \\ b & d \end{vmatrix}=0.$$

\item  证明：复习行列式的基本性质。


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}[fragile=singleslide]{3.1.1. }
\begin{frame}{4.6. 习题3 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  习题3：在空间直角坐标系中，记 $O=(0,0,0)$ 为原点。设向量 $\overset{\longrightarrow}{OA}=(x_1,y_1,z_1)$, 
$\overset{\longrightarrow}{OB}=(x_2,y_2,z_2)$, $\overset{\longrightarrow}{OC}=(x_3,y_3,z_3)$. 证明：以这三个向量为棱的平行六面体的体积为下述行列式的值，
$$\begin{vmatrix} x_1 & x_2 & x_3 \\  y_1 & y_2 & y_3 \\ z_1 & z_2 & z_3 \\  \end{vmatrix}. $$

\item  解答思路：
\begin{itemize}
\item  先考虑对角线之外都是零的情形。
\item  再考虑上三角行列式的情形。
\item  最后考虑一般的情形。
\end{itemize}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}[fragile=singleslide]{3.1.1. }
\begin{frame}{4.7. 习题4 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  习题4：在空间直角坐标系中，记 $O=(0,0,0)$ 为原点。设向量 $\overset{\longrightarrow}{OA}=(x_1,y_1,z_1)$, 
$\overset{\longrightarrow}{OB}=(x_2,y_2,z_2)$, $\overset{\longrightarrow}{OC}=(x_3,y_3,z_3)$. 证明：这三个向量在同一个平面上，当且仅当
$$\begin{vmatrix} x_1 & x_2 & x_3 \\  y_1 & y_2 & y_3 \\ z_1 & z_2 & z_3 \\  \end{vmatrix}=0. $$

\item  思路：如果三个向量在同一个平面，那么其中一个向量可以写成另外两个向量的组合。这类似于行列式的第三类初等变换。


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}[fragile=singleslide]{3.1.1. }
\begin{frame}{4.8. 行列式的子式，习题5 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  {\color{red}定义：设 $D$ 是 $n$ 阶行列式。取定其中的 $k$ 行与 $k$ 列，这些行与列的十字路口所在的元素组成一个 $k$ 阶的行列式，称为 $D$ 的一个 $k$ 阶子式。}

\item  习题5：写出下述行列式的所有取值不为零的二阶子式，
{\footnotesize  
\begin{eqnarray*}
D = \begin{vmatrix} 
1&2&0&0 \\
0&0&3&4 \\
5&6&0&0 \\
0&0&7&8 \\
\end{vmatrix}.
\end{eqnarray*}
}

\item  答案：一共有 18个取值不为零的二阶子式。

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}[fragile=singleslide]{3.1.1. }
\begin{frame}{4.9. 习题6 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  习题6：在下述五阶行列式中，先取定第1, 2行，再取第 $j, k$ 列，
{\footnotesize  
\begin{eqnarray*}
D = \begin{vmatrix} 
{\color{red}a_{11}}&a_{12}&{\color{red}a_{13}}&a_{14}& a_{15} \\ 
{\color{red}a_{21}} &a_{22}&{\color{red}a_{23}}&a_{24}&a_{25} \\ 
a_{31} &{\color{blue}a_{32}}&a_{33}&{\color{blue}a_{34}}&{\color{blue}a_{35}} \\ 
a_{41} &{\color{blue}a_{42}}&a_{43}&{\color{blue}a_{44}}&{\color{blue}a_{45}} \\ 
a_{51} &{\color{blue}a_{52}}&a_{53}&{\color{blue}a_{54}}&{\color{blue}a_{55}} \\ 
\end{vmatrix}
\end{eqnarray*}
}
得到的二阶行列式记为 $B_{1,2,j,k}$. 然后划去第1, 2行，再划去第 $j, k$ 列，剩下的三阶行列式记为 $M_{1,2,j,k}$. 又记 $A_{1,2,j,k} = (-1)^{1+2+j+k}M_{1,2,j,k}$. 最后记集合 
{\footnotesize  $S=\{(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)\}$. }
%\begin{enumerate}
%\item  
写出一些特殊的例子，验证等式 $D=\sum\limits_{(j,k)\in S} B_{1,2,j,k}A_{1,2,j,k}$ 是否成立。
%\item  证明在一般情形，上述等式也是成立的。
%\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}[fragile=singleslide]{3.1.1. }
\begin{frame}{4.10. 排列的奇偶性，习题7 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  {\color{red}定义：数字 $1,2,3,\cdots,n$ 的一个排列 $\sigma=(i_1,i_2,i_3,\cdots,i_n)$ 的逆序数 $\pi(\sigma)$ 是指这个排列的所有逆序的个数，其中若有 $s<t$ 但是 $i_s>i_t$ 则称 $(i_s,i_t)$ 是这个排列的一个逆序。}

\item  {\color{red}定义：如果一个排列 $\sigma = (i_1,i_2,i_3,\cdots,i_n)$ 的逆序数 $\pi(\sigma)$ 是奇（偶）数，那么称这个一个奇（偶）排列。}

\item  习题7：
\begin{enumerate}
\item  判断排列 $\sigma=(6,5,4,3,2,1)$ 和排列 $\tau=(7,6,5,4,3,2,1)$ 的奇偶性。
\item  证明：对换一个排列的任意两个数字，总是改变这个排列的奇偶性。
\end{enumerate}

\item  答案：
\begin{enumerate}
\item  两个都是奇排列。
\item  设对换的数字是 $i_s$ 与 $i_t$. 考察排列 $(i_1, \cdots, i_s, \cdots, i_k, \cdots, i_t, \cdots, i_n)$ 与排列 $(i_1, \cdots, i_t, \cdots, i_k, \cdots, i_s, \cdots, i_n)$ 中的数对的逆序的变化。
\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}[fragile=singleslide]{3.1.1. }
\begin{frame}{4.11. 习题8 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  习题8：将一副扑克牌的同一花色的13张牌随机放入 $4\times 4$ 的格子中，每个格子放一张牌，有三个剩余的空格。如果一张牌的旁边（上下左右）有空格，那么这张牌可以移动到这个空格里。是否可以找到若干次移动，从左图变成右图？如果三个空格的后两格不能放牌，情况又如何？

\begin{figure}
\centering
\includegraphics[height=0.3\textheight, width=0.5\textwidth]{4-det-permutation.png}
% \caption{ }
\end{figure}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}


